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Properties

Label 2-3311-3311.1679-c0-0-1
Degree $2$
Conductor $3311$
Sign $0.762 - 0.647i$
Analytic cond. $1.65240$
Root an. cond. $1.28545$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.95 + 0.355i)2-s + (2.77 + 1.04i)4-s + (−0.309 − 0.951i)7-s + (3.36 + 2.01i)8-s + (0.0448 + 0.998i)9-s + (−0.983 + 0.178i)11-s + (−0.267 − 1.97i)14-s + (3.64 + 3.18i)16-s + (−0.267 + 1.97i)18-s − 1.99·22-s + (−1.07 − 1.34i)23-s + (0.691 − 0.722i)25-s + (0.133 − 2.96i)28-s + (0.0620 + 0.0648i)29-s + (3.56 + 4.47i)32-s + ⋯
L(s)  = 1  + (1.95 + 0.355i)2-s + (2.77 + 1.04i)4-s + (−0.309 − 0.951i)7-s + (3.36 + 2.01i)8-s + (0.0448 + 0.998i)9-s + (−0.983 + 0.178i)11-s + (−0.267 − 1.97i)14-s + (3.64 + 3.18i)16-s + (−0.267 + 1.97i)18-s − 1.99·22-s + (−1.07 − 1.34i)23-s + (0.691 − 0.722i)25-s + (0.133 − 2.96i)28-s + (0.0620 + 0.0648i)29-s + (3.56 + 4.47i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $0.762 - 0.647i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (1679, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ 0.762 - 0.647i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.116852374\)
\(L(\frac12)\) \(\approx\) \(4.116852374\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.983 - 0.178i)T \)
43 \( 1 + (0.963 - 0.266i)T \)
good2 \( 1 + (-1.95 - 0.355i)T + (0.936 + 0.351i)T^{2} \)
3 \( 1 + (-0.0448 - 0.998i)T^{2} \)
5 \( 1 + (-0.691 + 0.722i)T^{2} \)
13 \( 1 + (0.134 - 0.990i)T^{2} \)
17 \( 1 + (0.134 + 0.990i)T^{2} \)
19 \( 1 + (0.393 - 0.919i)T^{2} \)
23 \( 1 + (1.07 + 1.34i)T + (-0.222 + 0.974i)T^{2} \)
29 \( 1 + (-0.0620 - 0.0648i)T + (-0.0448 + 0.998i)T^{2} \)
31 \( 1 + (-0.936 - 0.351i)T^{2} \)
37 \( 1 + (0.975 - 0.316i)T + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.963 + 0.266i)T^{2} \)
47 \( 1 + (0.393 - 0.919i)T^{2} \)
53 \( 1 + (-0.308 + 0.722i)T + (-0.691 - 0.722i)T^{2} \)
59 \( 1 + (-0.473 + 0.880i)T^{2} \)
61 \( 1 + (-0.936 + 0.351i)T^{2} \)
67 \( 1 + (-1.22 + 1.53i)T + (-0.222 - 0.974i)T^{2} \)
71 \( 1 + (1.17 + 0.0527i)T + (0.995 + 0.0896i)T^{2} \)
73 \( 1 + (-0.983 + 0.178i)T^{2} \)
79 \( 1 + (0.312 - 0.430i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.936 - 0.351i)T^{2} \)
89 \( 1 + (0.623 + 0.781i)T^{2} \)
97 \( 1 + (0.995 - 0.0896i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.280238326574734285873057854349, −7.957124756121289866543953332822, −7.06374438914402825280438437979, −6.59977838231612026500088980111, −5.71922713694643787916683012209, −4.84809575091191870574644913325, −4.52413226127472136725959928696, −3.56094767556591077200995423960, −2.70713858998201516937551567022, −1.91685885349350417917155188708, 1.58277331305716894234500385741, 2.58813090829945481994309168498, 3.29911394169635219283690422400, 3.91006577260224189576724116242, 5.03699789194437066477595200405, 5.56493844332525859563318359839, 6.10814611768759568206401644808, 6.91406352854594427989825340690, 7.61161269835221619854372961621, 8.735297765987525590532158444440

Graph of the $Z$-function along the critical line